We turn now to what is arguably one of the least well behaved modal languages ever proposed: first-order modal logic. However, in one of those twists that make intellectual history so fasci- nating, first-order modal logic has come to be accepted (at least in philosophical quarters) as the most important modal logic of all. For many philosophers, modal logic is first-order modal logic. This is not to say that first-order modal logic is philosophically uncontroversial. Indeed, as is discussed in Chapter 21 of this handbook, one of the liveliest debates in 20th century analytic philosophy was ignited when Quine [96] questioned the coherence of the enterprise. But two advances lead to its acceptance. The first was the development of the relational semantics of first-order modal logic (Kripke [75, 77] are key papers here) and the second was the publication of “Naming and Necessity” (Kripke [78]) which presented what is probably the most widely accepted philosophical interpretation of the technical machinery. While these developments did not dispel all the controversy, nowadays first-order modal logic together with (some form of) relational semantics, is generally regarded as a well understood (perhaps even boringly familiar) tool of philosophical analysis.
Viewed from a mathematical perspective, however, things look rather different. Had first- order modal logic never existed, a logician who proposed its (now standard) syntax and relational semantics might have been regarded as audacious, perhaps downright careless. Why? Because,
in essence, first-order modal logic is a combined logic. As we have just seen, combining two modal logics while retaining interesting properties is no easy matter. So it should not come as too much of a surprise that combining propositional modal logic with first-order logic is unlikely to be plain sailing. In what follows we shall sketch the standard syntax and semantics, and mention some of its problematic features.
First the syntax (we omit some of the clauses for the booleans):
ϕ ::= P(x1, . . . , xn)|x=y| ¬ϕ|ϕ→ψ|3ϕ|2ϕ| ∃xϕ| ∀xϕ.
HerePis ann-place predicate symbol and thexiare individual variables. So (given the clauses for the quantifiers and booleans) it is clear that we have a full first-order language at our disposal, and hence (because of the presence of the modalities) we can now search for first-order informa- tion at accessible states in the familiar way. But we can do more. The clauses for the quantifiers hide a subtlety: if a formulaϕcontains free first-order variables within the scope of a modality, then formulas of the form∀xϕand∃xϕbind variables within the scope of the modality. This possibility is what lead to Quine’s philosophical objections (“no binding into intensional con- texts”). And from a technical perspective it means we are combining two very different styles of logic in a way that allows a strong form of interaction.
The standard semantics for first-order modal logic comes in a number of variant forms. One basic choice concerns the domain of quantification: should the quantifiers range over some fixed domain of quantification (the constant domain semantics), or should each point should associated with its own domain (the varying domain semantics)? Here we shall present the varying domain semantics; for a discussion of the constant domain approaches, and of equivalences between the constant domain, varying domain, and other approaches, see Chapter 9 of this handbook, or Fitting and Mendelsohn [43].
DEFINITION 48. A varying domain model is a tuple(W, R, D,{δw}w∈W,{Vw}w∈W). Here
W is a non-empty set;Ris a binary relation onW;D(the domain of quantification) is a non- empty set; for allw ∈ W,δw ⊆ D; and for allw ∈ W,Vw is a function that assigns to each
n-place predicate symbol a subset ofDn.
That is, we have the familiar modal machinery from the propositional case (note that(W, R) is just a frame, and theVware essentially our familiar valuations upgraded to interpret first-order
n-place predicate symbolsP rather than propositional symbolsp) augmented by a specification (theδw) of the individuals the quantifiers at each statewrange over. We interpret first-order modal logic by taking such a model, together with an assignment of values to variables (that is, a functiongthat maps the individual variables to elements ofD), and using the following satisfaction definition:
M, g, w|=P(x1, . . . , xn) iff (g(x1), . . . g(xn))∈Vw(P),
M, g, w|=x=y iff g(x) =g(y), M, g, w|=¬ϕ iff notM, g, w|=ϕ,
M, g, w|=ϕ→ψ iff M, g, w6|=ϕ or M, g, w|=ψ,
M, g, w|=3ϕ iff for somev∈W such thatRwvwe haveM, g, v|=ϕ, M, g, w|=2ϕ iff for allv∈W such thatRwvwe haveM, g, v|=ϕ,
M, g, w|=∃ϕ iff for someg0∼
xgwhereg0(x)∈δwwe haveM, g0, v|=ϕ,
(Hereg0 ∼xgmeans that the assignmentsgandg0 are identical save possibly in the value they assign to the variablex.)
This language is capable of expressing some important distinctions. Consider, for example, the formulas∀x2ϕand2∀xϕ. The first asserts, of each existing entity, that it has the property
ϕat all accessible states. The second asserts that, at each accessible state, each entity that exists at that particular state has propertyϕ. Should either of these formulas imply the other? That is, should we accept as valid either of the following two principles?
∀x2ϕ→2∀xϕ Barcan formula
2∀xϕ→ ∀x2ϕ Converse Barcan formula
Instead of trying to answer such tricky philosophical questions (which bear on the de dicto/de re distinction, discussed in Chapter 9 of this handbook) let us consider what they say in the light of the relational interpretation just given. It is not difficult to see that the Barcan formula is valid in a varying domain model iff that model has decreasing domains, that is, if for allw, v∈W,Rwv
impliesδv ⊆ δw. And the Converse Barcan formula is valid on precisely increasing domain models, that is, models with the property thatRwvimpliesδw ⊆δv. So to insist on the validity of both principles is to force an even stronger interaction between the quantifiers and modalities: it takes us to a locally constant domain semantics in whichRwvimpliesδw=δv. This is a good example of the clarity that relational semantics can bring to difficult conceptual issues, and shows why first-order modal logic can be useful in philosophical logic and natural language semantics. So what’s the problem? Simply this: for all its analytical utility, first-order modal logic under its standard semantics is not well behaved mathematically. Early signs of trouble appeared in Fine [39], which showed that interpolation and the Beth property fail for first-order S5 under the varying domain semantics, and for any first-order modal logic between K and S5 under the constant domain semantics. As S5 is both philosophically central (it is often taken as to be embody the logic of “necessarily” and “possibly”) and semantically extremely straightforward (it is the logic of frames in whichRis an equivalence relation) these are strong negative results indeed. Worse was to come. It turns out that it is possible to take a propositional modal logic that is complete with respect to some class of frames, axiomatically extend it in the manner naturally suggested by the standard semantics, and yet to wind up with an incomplete first-order modal logic (see Ghilardi [50], Shehtman and Skvortsov [104], Corsi and Ghilardi [24], Cresswell [25]). Now, the issue here is not so much the incompleteness in itself (as we have already discussed, even in the propositional modal logic, frame incompleteness results are the norm) rather it is the
loss of completeness in the transition from the propositional case to the first-order case that is
worrying. To use the terminology introduced when we discussed combinations of logics: the standard relational semantics for first-order logic is a method of combination for which transfer of completeness fails.
Such results have led to renewed technical interest in first-order modal logic. The semantics of first-order modal logic has come under intense scrutiny, and a number of alternative seman- tics have been proposed which enable completeness results to be transferred. Some of this work has been model-theoretic (see, in particular, van Benthem’s [120] use of functional frames) but most of it has been highly abstract, employing the language of category theory; for a detailed account of such work, see Chapter 9 of this handbook. More recently, the hybrid logic com- munity has pointed out that upgrading the underlying propositional modal language to a hybrid language is another way to repair the situation: interpolation is regained (see Areces, Blackburn and Marx [7]), indeed, regained constructively (see Blackburn and Marx [7]) and general pos- itive results on transfer of completeness can be proved (see Blackburn and Marx [14]). All in
all, first-order modal logic is one of the most intriguing areas of modal logic: the most venerable system of all poses some of the deepest question about what it is to be truly modal.